Problem 18 · 2024 Math Kangaroo
Hard
Algebra & Patterns
evaluate-formulasubstitution
Jean-Philippe has \(n^3\) equally sized cubes. He uses them to build one big cube and paints its surface. The number of small cubes with exactly one painted face is then the same as the number of small cubes with no painted face. What is the value of n?
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Answer: D — 8
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Hint 1 of 2
For an n x n x n cube, count edge cubes (one painted face) and interior cubes (no painted face).
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Hint 2 of 2
Set the one-face count equal to the zero-face count and solve.
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Approach: equate the one-face and zero-face cube counts
- Cubes with exactly one painted face number 6(n-2)^2; cubes with no painted face number (n-2)^3.
- Setting 6(n-2)^2 = (n-2)^3 gives n - 2 = 6.
- So n = 8.
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